Correct!
If what you mean is: by trying to push point p with your finger you can tell if the system has a lot of mass (hence inertia) or not, then yes - this is correct.
Not sure what you mean by "additional signal" - please explain. The tension in the string is the only "signal" that is transmitted. It takes more tension to move the system when it has a lot of mass than if it has a small amount of mass.
Don't forget that the difference between F and R scenarios is that the F scenario adds a lot of mass to the system. You can definitely detect this mass when you try to accelerate it.
What you may not be grassping is the fact that although at rest both systems have 98 N of tension in the string, they have different tensions when you push on p to accelerate the system at 4.9 m/s^2. In the R scenario you push on p with .0147 N of force to the right - this increases the tension to the left of p by 0.0049 N, and decreases the tension to the right side by 0.0049 N, hence making an unbalanced force at R1 and R2 of 0.0049N to the right, which moves both R's to the right at 4.9 m/s^2. But in the F scenario, you have to push with 98.0049 N of force, which increases the tension to the left by 49 N and decreases the tension to the right by 49 N. This in turn causes unbalanced forces at F1 and F2 that are enough to accelerate both obectcts are 4.9 m/s^2.
Hope this helps.

